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Difficulty: MediumEquilibrium of Forces, Center of Gravity and Moments

A uniform horizontal wooden rod XYXY of length 3.0 m3.0\text{ m} and weight 50 N50\text{ N} rests horizontally on two smooth supports located at XX (the left end) and at a point ZZ which is 0.6 m0.6\text{ m} from end YY. A block of weight 120 N120\text{ N} is placed on the rod at a distance of 0.9 m0.9\text{ m} from end XX. What is the magnitude of the upward reaction force, in newtons, at support ZZ?

Answer: 76.25 N

Answer

The magnitude of the upward reaction force at support ZZ is 76.25 N76.25\text{ N}.
Taking moments about support XX, the total clockwise moment is the sum of the moment due to the load (120 N×0.9 m=108 N⋅m120\text{ N} \times 0.9\text{ m} = 108\text{ N}\cdot\text{m}) and the weight of the rod (50 N×1.5 m=75 N⋅m50\text{ N} \times 1.5\text{ m} = 75\text{ N}\cdot\text{m}), giving 183 N⋅m183\text{ N}\cdot\text{m}. Equating this to the counterclockwise moment of the reaction force at ZZ (RZ×2.4 mR_Z \times 2.4\text{ m}) yields RZ=1832.4=76.25 NR_Z = \frac{183}{2.4} = 76.25\text{ N}.

Step-by-Step Solution

1
Identify the perpendicular distance of each force and support from pivot point XX.
Center of gravity position xcg=1.5 mx_{cg} = 1.5\text{ m}, load position xL=0.9 mx_{L} = 0.9\text{ m}, and support ZZ position xZ=3.0−0.6=2.4 mx_{Z} = 3.0 - 0.6 = 2.4\text{ m}.
Taking moments about XX requires knowing the exact moment arm for each force from XX.
2
Set up the equation for rotational equilibrium about point XX.
Total clockwise moment = (120×0.9)+(50×1.5)=183 N⋅m(120 \times 0.9) + (50 \times 1.5) = 183\text{ N}\cdot\text{m}; Total counterclockwise moment = RZ×2.4R_Z \times 2.4.
Choosing pivot XX eliminates the unknown reaction force RXR_X because its distance from XX is zero.
3
Equate clockwise moments to counterclockwise moments and solve for RZR_Z.
RZ=1832.4=76.25 NR_Z = \frac{183}{2.4} = 76.25\text{ N}.
For a body in rotational equilibrium, the algebraic sum of moments about any point must equal zero.

Key Concept

Principle of Moments and Rotational Equilibrium
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Topics

Equilibrium of Forces, Center of Gravity and Moments

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